237. If two circles are tangent internally, all chords of the greater circle drawn from the point of contact are divided proportionally by the circumference of the smaller circle. HINT. Draw any two of the chords, join the points where they meet the circumferences, and prove that the A thus formed are similar. 238. In an inscribed quadrilateral, the product of the diagonals is equal to the sum of the products of the opposite sides. HINT. Draw DE, making ▲ CDE=LADB. The AABD and CDE are similar. Also the ABCD and ADE are similar. 239. The sum of the squares of the four sides of any quadrilateral is equal to the sum of the squares of the diagonals increased by four times the square of the line joining the middle points of the diagonals. HINT. Join the middle points F, E, of the diagonals. Draw EB and ED. Apply & 344 to the AABC and ADC, add the results, and eliminate BE2 + DE2 by applying ? 343 to the ▲ BDE. D D E B 240. The square of the bisector of an exterior angle of a triangle is equal to the product of the external segments deter mined by the bisector upon one of the sides dimin ished by the product of the other two sides. F H B D HINT. Let CD bisect the exterior BCH of the AABC. Circumscribe a O about the A, produce DC to meet the circumference in F, and draw BF. Prove & ACD, BCF similar. Apply 8 347. 241. If a point O is joined to the vertices of a triangle ABC, and through any point A'in OA a line parallel to AB is drawn, meeting OB at B', and then through B′ a line parallel to BC, meeting OC at C′, and C is joined to A', the triangle A'B'C' will be similar to the triangle ABC. 242. If the line of centres of two circles meets the circumferences at the points A, B, C, D, and meets the common exterior tangent at P, then PA× PD = PB × PC. 243. The line of centres of two circles meets the common exterior tangent at P, and a secant is drawn from P, cutting the circles at the consecutive points E, F, G, H. Prove that PE× PH = PF× PG. NUMERICAL EXERCISES. 244. A line is drawn parallel to a side AB of a triangle ABC, and cutting AC in D, BC in E. If AD: DC= 2: 3, and AB = 20 inches, find DE. 245. The sides of a triangle are 9, 12, 15. Find the segments made by bisecting the angles. ( 313.) 246. A tree casts a shadow 90 feet long, when a vertical rod 6 feet high casts a shadow 4 feet long. How high is the tree? 247. The bases of a trapezoid are represented by a, b, and the altitude by h. Find the altitudes of the two triangles formed by producing the legs till they meet. 248. The sides of a triangle are 6, 7, 8. In a similar triangle the side homologous to 8 is equal to 40. Find the other two sides. 249. The perimeters of two similar polygons are 200 feet and 300 feet. If a side of the first polygon is 24 feet, find the homologous side of the second polygon. 250. How long must a ladder be to reach a window 24 feet high, if the lower end of the ladder is 10 feet from the side of the house? 251. If the side of an equilateral triangle = a, find the altitude. 252. If the altitude of an equilateral triangle = h, find the side. 253. Find the lengths of the longest and the shortest chords that can be drawn through a point 6 inches from the centre of a circle whose radius is equal to 10 inches. 254. The distance from the centre of a circle to a chord 10 inches long is 12 inches. Find the distance from the centre to a chord 24 inches long. 255. The radius of a circle is 5 inches. Through a point 3 inches from the centre a diameter is drawn, and also a chord perpendicular to the diameter. Find the length of this chord, and the distance from one end of the chord to the ends of the diameter. 256. The radius of a circle is 6 inches. Through a point 10 inches from the centre tangents are drawn. Find the lengths of the tangents, and also of the chord joining the points of contact. 257. If a chord 8 inches long is 3 inches distant from the centre of the circle, find the radius and the distances from the end of the chord to the ends of the diameter which bisects the chord. 258. The radius of a circle is 13 inches. Through a point 5 inches from the centre any chord is drawn. What is the product of the two segments of the chord? What is the length of the shortest chord that can be drawn through the point? 259. From the end of a tangent 20 inches long a secant is drawn through the centre of the circle. If the exterior segment of this secant is 8 inches, find the radius of the circle. 260. The radius of a circle is 9 inches; the length of a tangent is 12 inches. Find the length of a secant drawn from the extremity of the tangent to the centre of the circle. 261. The radii of two circles are 8 inches and 3 inches, and the distance between their centres is 15 inches. Find the lengths of their common tangents. 262. Find the segments of a line 10 inches long divided in extreme and mean ratio. 263. The sides of a triangle are 4, 5, 6. Is the largest angle acute, right, or obtuse ? PROBLEMS. 264. To divide one side of a given triangle into segments proportional to the adjacent sides. (8 313.) 265. To produce a line AB to a point C'so that AB: AC-3: 5. 266. To find in one side of a given triangle a point whose distances from the other sides shall be to each other in a given ratio. 267. Given an obtuse triangle; to draw a line from the vertex of the obtuse angle to the opposite side which shall be a mean proportional between the segments of that side. 268. Through a given point P within a given circle to draw a chord AB so that AP: BP-2: 3. 269. To draw through a given point P in the arc subtended by a chord AB a chord which shall be bisected by AB. 270. To draw through a point P, exterior to a given circle, a secant PAB so that PA: AB4: 3. 271. To draw through a point P, exterior to a given circle, a secant PAB so that AB2 PAX PB. = 272. To find a point P in the arc subtended by a given chord AB so that PA: PB = 3: 1. 273. To draw through one of the points of intersection of two circles a secant so that the two chords that are formed shall be to each other in the ratio of 3: 5. 274. To divide a line into three parts proportional to 2, †, §. 275. Having given the greater segment of a line divided in extreme and mean ratio, to construct the line. 276. To construct a circle which shall pass through two given points and touch a given straight line. 277. To construct a circle which shall pass through a given point and touch two given straight lines. 278. To inscribe a square in a semicircle. 279. To inscribe a square in a given triangle. HINT. Suppose the problem solved, and DEFG the inscribed square. Draw CM || to AB, and let AF produced meet CM in M. Draw CH and MN 1 to AB, and produce AB to meet MN at N. The ACM, AGF are similar; also the ▲ AMN, AFE are similar. By these triangles show that the figure CMNH is a square. By constructing this square, the point F can be found. G C ADHE B M 280. To inscribe in a given triangle a rectangle similar to a given rectangle. 281. To inscribe in a circle a triangle similar to a given triangle. 282. To inscribe in a given semicircle a rectangle similar to a given rectangle. Find the locus of a point the distances of which from two given points are in a given ratio. 283. To circumscribe about a circle a triangle similar to a given triangle. 284. To construct the expression, x = 2 abc de 2 ab с ; that is, d e 285. To construct two straight lines, having given their sum and their ratio. 286. To construct two straight lines, having given their difference and their ratio. 287. Having given two circles, with centres O and O', and a point A in their plane, to draw through the point A a straight line, meeting the circumferences at B and C, so that AB: AC = 1:2. HINT. Suppose the problem solved, join OA and produce it to D, making OA: AD=1:2. Join DC; & OAB, ADC are similar. BOOK IV. AREAS OF POLYGONS. refered 358. The area of a surface is the ratio of the surface to the unit of surface. The unit of surface is a square whose side is a unit of length; as the square inch, the square foot, etc. 359. Equivalent figures are figures having equal areas. PROPOSITION I. THEOREM. 360. The areas of two rectangles having equal altitudes are to each other as their bases. Let the two rectangles be AC and AF, having the same altitude AD. Proof. CASE I. When AB and AE are commensurable. Suppose AB and AE have a common measure, as AO, which is contained in AB seven times and in AE four times. Apply this measure to AB and AE, and at the several points of division erect Is. The rect. AC will be divided into seven rectangles, and the rect. AF will be divided into four rectangles. |